BPMN and Beyond Business process modelling notation, workflow ...

and refer the interested reader for this to the AsmBook [6]. To define the various interpretations of
the OR-join as different instantiations of one abstract model we make use of the ASM refinement
method defined in [3].
3.2 Business Process Diagrams
As common in the field, we mathematically represent any business process as a graph. The nodes
represent the workflow objects, where activities are performed depending on a) resources being
available, b) data or control conditions to be true and c) events to happen, as described by transition
rules associated to nodes. These rules define the meaning of the corresponding workflow constructs.
The arcs define the graph traversal, i.e. the order in which the workflow objects are visited for the
execution of the associated rules.
We freely use the usual graph-theoretic concepts, for example source(arc), target(arc) for source
and target node of an arc, pred(node) for the set of source nodes of arcs that have the given node
as target node, inArc(node) for the set of arcs with node as target node, similarly succ(node) for
the set of target nodes of arcs that have the given node as source node, outArc(node) for the set
of arcs with node as source node, etc.
All the workflow transition rules, associated to nodes to describe the meaning of the workflow
construct associated to this node, take the following form (usually instantiated by additional parameters).
They state upon which events and under which further conditions on the control flow,
the underlying data and the availability of resources, the rule can fire to perform specific operations
on the underlying data (‘how to change the internal state’) and control (‘where to proceed’), to
possibly trigger new events (besides consuming the triggering ones) and to operate on the resource
space to take possession of the needed (or to release not any more needed) resources.
WorkflowTransition(node) =
if EventCond(node) and CtlCond(node)
and DataCond(node) and ResourceCond(node) then
DataOp(node)
CtlOp(node)
EventOp(node)
ResourceOp(node)
A workflow or business process modeling language interpreter is a set of such rules, covering all
language constructs, together with a scheduler to choose at each moment a node where a rule can
be fired, which is the case when its guard is true in the current state. In this way one can define
for example the semantics of the BPMN standard by an interpreter with rules (more precisely rule
schemes) for each BPMN flow object (activities, events, gateways) [7]. For the discussion of the
OR-join problem we can focus the discussion on gateways only (see Sect. 3.4). Furthermore, for
this discussion events and resources play no role and therefore will not be mentioned any more.
3.3 Token-Based Sequence Flow Interpretation
Although the BPMN standard document declares to use the token-based interpretation of control
flow only for illustrative purposes [8, p.35], for the sake of definiteness we represent it mathematically
by associating tokens—elements of a set Token—to arcs, using a dynamic function
token(arc). 8 A token typically includes information on (the processID of) the process instance to
which it belongs. Typically token(arc) denotes a multiset of tokens currently residing on arc.
token : Arc → Multiset(Token)
8 We deliberately avoid introducing yet another category of graph items, like the so-called places in Petri
nets, whose only role would be to hold these tokens.
6